[{"publication_identifier":{"eissn":["2673-8716"]},"has_accepted_license":"1","publication_status":"published","intvolume":"         5","citation":{"ieee":"M. T. Meyer and A. Schindlmayr, “Generalized Miller formulae for quantum anharmonic oscillators,” <i>Dynamics</i>, vol. 5, no. 3, Art. no. 34, 2025, doi: <a href=\"https://doi.org/10.3390/dynamics5030034\">10.3390/dynamics5030034</a>.","chicago":"Meyer, Maximilian Tim, and Arno Schindlmayr. “Generalized Miller Formulae for Quantum Anharmonic Oscillators.” <i>Dynamics</i> 5, no. 3 (2025). <a href=\"https://doi.org/10.3390/dynamics5030034\">https://doi.org/10.3390/dynamics5030034</a>.","ama":"Meyer MT, Schindlmayr A. Generalized Miller formulae for quantum anharmonic oscillators. <i>Dynamics</i>. 2025;5(3). doi:<a href=\"https://doi.org/10.3390/dynamics5030034\">10.3390/dynamics5030034</a>","mla":"Meyer, Maximilian Tim, and Arno Schindlmayr. “Generalized Miller Formulae for Quantum Anharmonic Oscillators.” <i>Dynamics</i>, vol. 5, no. 3, 34, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/dynamics5030034\">10.3390/dynamics5030034</a>.","short":"M.T. Meyer, A. Schindlmayr, Dynamics 5 (2025).","bibtex":"@article{Meyer_Schindlmayr_2025, title={Generalized Miller formulae for quantum anharmonic oscillators}, volume={5}, DOI={<a href=\"https://doi.org/10.3390/dynamics5030034\">10.3390/dynamics5030034</a>}, number={334}, journal={Dynamics}, publisher={MDPI}, author={Meyer, Maximilian Tim and Schindlmayr, Arno}, year={2025} }","apa":"Meyer, M. T., &#38; Schindlmayr, A. (2025). Generalized Miller formulae for quantum anharmonic oscillators. <i>Dynamics</i>, <i>5</i>(3), Article 34. <a href=\"https://doi.org/10.3390/dynamics5030034\">https://doi.org/10.3390/dynamics5030034</a>"},"oa":"1","date_updated":"2025-10-10T07:29:36Z","volume":5,"author":[{"first_name":"Maximilian Tim","orcid":"0009-0003-4899-0920","last_name":"Meyer","id":"77895","full_name":"Meyer, Maximilian Tim"},{"orcid":"0000-0002-4855-071X","last_name":"Schindlmayr","id":"458","full_name":"Schindlmayr, Arno","first_name":"Arno"}],"doi":"10.3390/dynamics5030034","type":"journal_article","status":"public","_id":"60959","department":[{"_id":"296"},{"_id":"230"},{"_id":"15"},{"_id":"170"},{"_id":"35"}],"user_id":"458","isi":"1","article_number":"34","article_type":"original","file_date_updated":"2025-08-28T12:27:05Z","quality_controlled":"1","issue":"3","year":"2025","publisher":"MDPI","date_created":"2025-08-20T09:46:13Z","title":"Generalized Miller formulae for quantum anharmonic oscillators","publication":"Dynamics","abstract":[{"text":"Miller's rule originated as an empirical relation between the nonlinear and linear optical coefficients of materials. It is now accepted as a useful tool for guiding experiments and computational materials discovery, but its theoretical foundation had long been limited to a derivation for the classical Lorentz model with a weak anharmonic perturbation. Recently, we developed a mathematical framework which enabled us to prove that Miller's rule is equally valid for quantum anharmonic oscillators, despite different dynamics due to zero-point fluctuations and further quantum-mechanical effects. However, our previous derivation applied only to one-dimensional oscillators and to the special case of second- and third-harmonic generation in a monochromatic electric field. Here we extend the proof to three-dimensional quantum anharmonic oscillators and also treat all orders of the nonlinear response to an arbitrary multi-frequency field. This makes the results applicable to a much larger range of physical systems and nonlinear optical processes. The obtained generalized Miller formulae rigorously express all tensor elements of the frequency-dependent nonlinear susceptibilities in terms of the linear susceptibility and thus allow a computationally inexpensive quantitative prediction of arbitrary parametric frequency-mixing processes from a small initial dataset.","lang":"eng"}],"file":[{"creator":"schindlm","date_created":"2025-08-28T12:23:26Z","date_updated":"2025-08-28T12:27:05Z","access_level":"open_access","file_name":"dynamics-05-00034.pdf","file_id":"61056","title":"Generalized Miller formulae for quantum anharmonic oscillators","file_size":375897,"description":"Creative Commons Attribution 4.0 International Public License (CC BY 4.0)","content_type":"application/pdf","relation":"main_file"}],"external_id":{"isi":["001581270200001"]},"ddc":["530"],"language":[{"iso":"eng"}]},{"year":"2024","quality_controlled":"1","issue":"9","title":"Derivation of Miller's rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator","publisher":"IOP Publishing","date_created":"2024-03-22T08:44:39Z","abstract":[{"lang":"eng","text":"Miller's rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller's rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller's rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller's rule, and we illustrate this point by means of a numerical example."}],"file":[{"content_type":"application/pdf","relation":"main_file","creator":"schindlm","date_created":"2024-04-04T09:24:22Z","date_updated":"2024-04-04T09:24:22Z","access_level":"open_access","file_name":"Meyer_2024_J._Phys._B _At._Mol._Opt._Phys._57_095001.pdf","file_id":"53204","title":"Derivation of Miller's rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator","file_size":358155,"description":"Creative Commons Attribution 4.0 International Public License (CC BY 4.0)"}],"publication":"Journal of Physics B: Atomic, Molecular and Optical Physics","ddc":["530"],"language":[{"iso":"eng"}],"external_id":{"isi":["001196678300001"]},"intvolume":"        57","citation":{"mla":"Meyer, Maximilian Tim, and Arno Schindlmayr. “Derivation of Miller’s Rule for the Nonlinear Optical Susceptibility of a Quantum Anharmonic Oscillator.” <i>Journal of Physics B: Atomic, Molecular and Optical Physics</i>, vol. 57, no. 9, 095001, IOP Publishing, 2024, doi:<a href=\"https://doi.org/10.1088/1361-6455/ad369c\">10.1088/1361-6455/ad369c</a>.","bibtex":"@article{Meyer_Schindlmayr_2024, title={Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator}, volume={57}, DOI={<a href=\"https://doi.org/10.1088/1361-6455/ad369c\">10.1088/1361-6455/ad369c</a>}, number={9095001}, journal={Journal of Physics B: Atomic, Molecular and Optical Physics}, publisher={IOP Publishing}, author={Meyer, Maximilian Tim and Schindlmayr, Arno}, year={2024} }","short":"M.T. Meyer, A. Schindlmayr, Journal of Physics B: Atomic, Molecular and Optical Physics 57 (2024).","apa":"Meyer, M. T., &#38; Schindlmayr, A. (2024). Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator. <i>Journal of Physics B: Atomic, Molecular and Optical Physics</i>, <i>57</i>(9), Article 095001. <a href=\"https://doi.org/10.1088/1361-6455/ad369c\">https://doi.org/10.1088/1361-6455/ad369c</a>","ama":"Meyer MT, Schindlmayr A. Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator. <i>Journal of Physics B: Atomic, Molecular and Optical Physics</i>. 2024;57(9). doi:<a href=\"https://doi.org/10.1088/1361-6455/ad369c\">10.1088/1361-6455/ad369c</a>","chicago":"Meyer, Maximilian Tim, and Arno Schindlmayr. “Derivation of Miller’s Rule for the Nonlinear Optical Susceptibility of a Quantum Anharmonic Oscillator.” <i>Journal of Physics B: Atomic, Molecular and Optical Physics</i> 57, no. 9 (2024). <a href=\"https://doi.org/10.1088/1361-6455/ad369c\">https://doi.org/10.1088/1361-6455/ad369c</a>.","ieee":"M. T. Meyer and A. Schindlmayr, “Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator,” <i>Journal of Physics B: Atomic, Molecular and Optical Physics</i>, vol. 57, no. 9, Art. no. 095001, 2024, doi: <a href=\"https://doi.org/10.1088/1361-6455/ad369c\">10.1088/1361-6455/ad369c</a>."},"publication_identifier":{"issn":["0953-4075"],"eissn":["1361-6455"]},"has_accepted_license":"1","publication_status":"published","doi":"10.1088/1361-6455/ad369c","oa":"1","date_updated":"2024-04-13T11:20:56Z","volume":57,"author":[{"full_name":"Meyer, Maximilian Tim","id":"77895","last_name":"Meyer","orcid":"0009-0003-4899-0920","first_name":"Maximilian Tim"},{"first_name":"Arno","id":"458","full_name":"Schindlmayr, Arno","orcid":"0000-0002-4855-071X","last_name":"Schindlmayr"}],"status":"public","type":"journal_article","article_number":"095001","article_type":"original","isi":"1","file_date_updated":"2024-04-04T09:24:22Z","_id":"52723","department":[{"_id":"296"},{"_id":"230"},{"_id":"15"},{"_id":"170"},{"_id":"35"}],"user_id":"458"}]